Estimation of remaining propellant of a spacecraft at the beginning of life (BOL) or at the end of the transfer orbit, i.e., injection into geostationary orbit, has been generally carried out using a bookkeeping method. More specifically, this method primarily models mass flowrate by estimating flow coefficients from liquid apogee motor (LAM) ground test data. These coefficients are determined using a regression analysis on data collected during LAM acceptance testing. The coefficients are then used in a set of equations which relate mass flowrate to supply pressure so that the only remaining unknowns are chamber pressure, oxidizer mass flowrate and fuel mass flowrate.
The three unknown parameters are then determined by an iterative process which starts with an initial guess for the oxidizer mass flowrate. One equation is used to determine an initial chamber pressure. A second equation is used to determine the corresponding fuel mass flowrate. This fuel mass flowrate is then used in a third equation to determine a second chamber pressure. The first and second chamber pressures are compared and if they are not equal within a certain tolerance, the process is repeated. The oxidizer and fuel mass flowrates corresponding to the converged solution are then used in conjunction with the total engine firing time to determine total propellant consumption during transfer orbit.
This technique has several disadvantages. A conservative analysis indicates that the uncertainty in mass flowrate associated with this technique is at best approximately 2%. The large uncertainty is mainly associated with the fact that the chamber pressure is not known. The three equations used in the method are based on both laminar and turbulent pressure drop relations. This is done for modeling convenience, but it is a physical impossibility. The parameters used in the bookkeeping method are unobservable.
As an example of the above problem, one known type of satellite features an integrated propulsion subsystem that utilizes the same set of propellant tanks for transfer orbit as for on-orbit stationkeeping. Approximately 80% (.about.3000 lb) of the total propellant is consumed during the transfer orbit. The above-noted modeling uncertainty of the mass flow through the transfer orbit LAM is estimated as 2-3% to account for all potential error sources from ground instrumentation during acceptance testing to in-flight telemetry. As noted above, the largest factor in causing this uncertainty is the fact that the chamber pressure of the satellite thruster is not directly known, but rather is inferred from inlet pressure correlations. Thus, because the modelling uncertainty requires the need to include for a 2-3% error factor in predicting remaining amount of propellant, this results in the need to reduce the propellant prediction by 60-90 lbs of propellant, or 1-1.5 years of satellite life.
Clearly, the bookkeeping prediction method has an inherent worst case "knowledge penalty." In other words, a spacecraft may indeed contain more propellant than conservative bookkeeping suggests, but known techniques simply are not able to provide a more accurate calculation. As a result, an operator must plan to deorbit a satellite when bookkeeping predicts 1-1.5 years of remaining life. Thus, a need exists for a method which more reliably predicts remaining propellant in a spacecraft after attaining transfer orbit.